chapter 18 the lognormal distribution. copyright © 2006 pearson addison-wesley. all rights...
TRANSCRIPT
Chapter 18
The Lognormal Distribution
Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 18-2
( ; , )x e
x
1
2
1
2
2
The Normal Distribution
• Normal distribution (or density)
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),;(),;( xx
x N~ ( , ) 2
z N~ ( , )0 1
N a e dxxa
( )
1
2
1
22
The Normal Distribution (cont’d)
• Normal density is symmetric:
• If a random variable x is normally distributed with mean and standard deviation,
• z is a random variable distributed standard normal:
• The value of the cumulative normal distribution function N(a) equals to the probability P of a number z drawn from the normal distribution to be less than a. [P(z<a)]
Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 18-4
The Normal Distribution (cont’d)
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The Normal Distribution (cont’d)
• The probability of a number drawn from the standard normal distribution will be between a and –a:
Prob (z < –a) = N(–a)
Prob (z < a) = N(a)
therefore
Prob (–a < z < a) =
N(a) – N(–a) = N(a) – [1 – N(a)] = 2·N(a) – 1
• Example: Prob (–0.3 < z < 0.3) = 2·0.6179 – 1 = 0.2358
Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 18-6
The Normal Distribution (cont’d)
• Converting a normal random variable to standard normal: If , then if
• And vice versa: If , then if
• Example 18.2: Suppose and then , and
z N~ ( , )0 1x N~ ( , ) 2 zx
x N~ ( , ) 2z N~ ( , )0 1 x z
x N~ ( , )3 5 z N~ ( , )0 1x
N 35
0 1~ ( , ) 3 5 3 25 z N~ ( , )
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The Normal Distribution (cont’d)
• The sum of normal random variables is also
where xi, i = 1,…,n, are n random variables, with mean E(xi) = i, variance Var(xi) =i
2, covariance Cov(xi,xj) = ij = ijij
i i i i i j ijj
n
i
n
i
n
i
nx N~ ,
1111
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The Lognormal Distribution
• A random variable x is lognormally distributed if ln(x) is normally distributed
If x is normal, and ln(y) = x (or y = ex), then y is lognormal If continuously compounded stock returns are normal then
the stock price is lognormally distributed
• Product of lognormal variables is lognormal
If x1 and x2 are normal, then y1=ex1 and y2=ex
2 are lognormal
The product of y1 and y2: y1 x y2 = ex1 x ex
2 = ex1+x
2
Since x1+x2 is normal, ex1+x
2 is lognormal
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The Lognormal Distribution (cont’d)
• The lognormal density function
where S0 is initial stock price, and ln(S/S0)~N(m,v2), S is future stock price, m is mean, and v is standard deviation of continuously compounded return
• If x ~ N(m,v2), then
g S m v SSv
e
S S m v
v( ; , , )
ln( ) [ln( ) . ]
0
1
2
0 51
2
0
2 2
E e ex m v( )
1
22
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The Lognormal Distribution (cont’d)
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A Lognormal Model of Stock Prices
• If the stock price St is lognormal, St / S0 = ex, where x, the continuously compounded return from 0 to t is normal
• If R(t, s) is the continuously compounded return from t to s, and, t0 < t1 < t2, then R(t0, t2) = R(t0, t1) + R(t1, t2)
• From 0 to T, E[R(0,T)] = nh , and Var[R(0,T)] = nh2
• If returns are iid, the mean and variance of the continuously compounded returns are proportional to time
Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 18-12
Prob( ) ( )S K N dt 2
A Lognormal Model of Stock Prices (cont’d)
• If we assume that
then
and therefore
• If current stock price is S0, the probability that the option will expire in the money, i.e.
where the expression contains , the true expected return on the stock in place of r, the risk-free rate
2( 0.5 )0
t t ztS S e
In(St / S0 ) ~ N[( 0.5 2 )t, 2t]2
0( / ) ( 0.5 )tIn S S t t z
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S S etL t t N p
0
1
222 1( ) ( / )
S S etU t t N p
0
1
222 1( ) ( / )
Lognormal Probability Calculations
• Prices StL and St
U such that Prob (StL < St ) = p/2 and
Prob (StU > St ) = p/2
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E S S K SeN d
N dt t
t( | )( )( )
( ) 1
2
P S K r t Ke N d e SN drt t( , , , , , ) ( ) ( ) 2 1
Lognormal Probability Calculations (cont’d)
• Given the option expires in the money, what is the expected stock price? The conditional expected price
where the expression contains a, the true expected return on the stock in place of r, the risk-free rate
• The Black-Scholes formula—the price of a call option on a nondividend-paying stock
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Estimating the Parameters of a Lognormal Distribution
• The lognormality assumption has two implications Over any time horizon continuously compounded
return is normal The mean and variance of returns grow proportionally
with time
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Estimating the Parameters of a Lognormal Distribution (cont’d)
• The mean of the second column is 0.006745 and the standard deviation is 0.038208
• Annualized standard deviation
• Annualized expected return
0.038208 52 0.2755
0.00674552 0.50.27550.2755 0.3877
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How Are Asset Prices Distributed?
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How Are Asset Prices Distributed? (cont’d)
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How Are Asset Prices Distributed? (cont’d)